Rainbow connection of bridgeless outerplanar graphs with small diameters

In this paper, we investigate rainbow connection number $rc(G)$ of bridgeless outerplanar graphs $G$ with diameter 2 or 3. We proved the following results: If $G$ has diameter $2,$ then $rc(G)=3$ for fan graphs $F_{n}$ with $n\geq 7$ or $C_5,$ otherwise $rc(G)=2;$ if $G$ has diameter $3,$ then $rc(G)\leq 4$ and the bound is sharp.